Time Optimal Control of the Brockett Integrator

The Brockett integrator is a 3-dimensional system with a non-holonomic constraint and the non-holonomic constraint makes the stabilization and control of the system challenging. The 3-dimensional non-holonomic integrator can be generalized in more than one way and this work looks into the time-optimal control of the generalized Brockett integrator which evolves on R × Skewm(R). The problem has been cast in Hamiltonian framework and the Pontryagin’s Maximum Principle has been used to obtain candidate control inputs for the time-optimal state transfer. The existence of solution to optimal control problem has been proved and a complete procedure is presented to achieve the state transfer in a time-optimal way. Abnormal extremals are shown to correspond to a stationary curve on the cotangent bundle which proves the non-existence of the abnormal extremals. Simulation results, which have been performed in MATLAB, are presented and the simulation results validate the results obtained theoretically.